Although these two sentences are correct individually, it may be worth it to emphasise that they should not be read together: the determinant is defined only for square matrices. (One can do something like computing the determinant of det(A^{adjoint}A) if you want a general numerical invariant that behaves as you like—and then we're getting in the neighbourhood of the SVD).
> If you take a 3 x 3 matrix and it transforms all the 3-D vectors into only planes (which are 2-D objects), the determinant will be zero.
Although your meaning seems clear, I think you may have misspoken slightly. Of course 3D vectors are transformed by a 3 x 3 matrix into vectors, not planes. To me, what I think you mean would read better as "If you take a 3 x 3 matrix and there is a plane such that all 3D vectors are transformed into that plane, then …", or, perhaps even better, "… such that the transforms of all 3D vectors lie in that plane …".